(see the discussion on [1]} preceding Remark (2.10)). Using these relations, it becomes possible to deduce that the coefficient of \(T^g\)  in \(f_{E_1,\Sigma _0}(T)\)  is \(p\) -adic unit if and only if that of \(T^g\)  in \(f_{E_2,\Sigma _0}(T)\)  is a \(p\) -adic unit (see the proofs of Corollory \(\ref {prethmcorollory}\)  and Theorem \(\ref {gammacongruence}\) ). By the results of Schneider and Perrin-Riou discussed earlier, the order of vanishing of \(f_{E_i}^{alg}(T)\)  at \(T=0\)  is \(g\) , and its leading coefficient is equal to \(\chi _t(\Gamma , E_i)\) , up to a \(p\) -adic unit. Setting \(\Phi _{E_i}:=\prod _{l\in \Sigma _0} |L_l(E_i,1)|_p\) , we then compare \(\Phi _{E_i}\times \chi _t(\Gamma , E_i)\)  for the two elliptic curves.
